Semiregular automorphisms of edge-transitive graphs

TL;DR

This paper investigates the existence of semiregular automorphisms in edge-transitive graphs, proving their existence in all regular edge-transitive graphs of valency three or four, thus contributing to the understanding of the polycirculant conjecture.

Contribution

It establishes that all regular edge-transitive graphs with valency three or four possess semiregular automorphisms, advancing the knowledge on automorphism structures in such graphs.

Findings

Regular edge-transitive graphs of valency three or four have semiregular automorphisms.

Supports the polycirculant conjecture for specific classes of graphs.

Provides new insights into automorphism group structures of symmetric graphs.

Abstract

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.